Number theory problems from the harmonic analysis of a fractal

We study some number theory problems related to the harmonic analysis (Fourier bases) of the Cantor set introduced by Jorgensen and Pedersen in \cite{JP98}

Tiling Properties Of Spectra Of Measures

We investigate tiling properties of spectra of measures, i.e., sets Λ in R such that {e 2πiλx : λ ∈ Λ} forms an orthogonal basis in L 2 (µ), where µ is some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard pairs in the case of Hadamard pairs of size 2,3,4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2,3,4 or

Tiling Properties Of Spectra Of Measures

We investigate tiling properties of spectra of measures, i.e., sets (Formula presented.) forms an orthogonal basis in (Formula presented.), where μ is some finite Borel measure on (Formula presented.). Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven–Meyerowitz property, the existence of complementing Hadamard pairs in the case of Hadamard pairs of size 2, 3, 4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2, 3, 4 or 5

Efficient Low-Error Analytical-Numerical Approximations For Radial Solutions Of Nonlinear Laplace Equations

We study radial solution of nonlinear elliptic partial differential equations of the form −△u=f(u) (a nonlinear Laplace equation) by means of an analytical-numerical method, namely optimal homotopy analysis. In this method, one obtain approximate analytical solutions which contain a free control parameter. This control parameter can be adjusted in order to improve the convergence or accuracy of the approximations. We outline the general technique for obtaining radial solutions of the general nonlinear elliptic partial differential equations of the form −△u=f(u), before focusing our attention on several specific equations, namely, the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind. For the general case, we outline the method by which one may control the residual errors of these analytical-numerical approximations. One benefit to this method is that one can obtain solutions with rather low residual errors after only a few terms in the analytical expansion are calculated. This makes the method rather efficient compared to a standard homotopy approach, where many terms may need to be computed to guarantee the accuracy of the solution. By studying the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind, we demonstrate that the benefits of the method are often related to the form of the nonlinearity inherent in the problem. For certain forms of nonlinearity, the method gives very accurate solutions after relatively few terms, while for other forms of nonlinearity this is not the case

Small-Time Existence And Two Classes Of Solutions For The N-Dimensional Coupled Yukawa Equations

We discuss the Yukawa equations, a system of nonlinear partial differential equations which has applications to meson-nucleon interactions. First, we determine small-time local existence of solutions under fairly general initial data. In particular, we show that small-time solutions are analytic, and by the higher order Cauchy–Kowalevski theorem, unique. Secondly, we obtain a class of stationary solutions. For the 1+1 model, we find that space-periodic solutions may be obtained through an application of multiple scales analysis. In this case, the coupled Yukawa equations result in a sort of non-local Gross-Pitaevskii equation. We outline the method for stationary solutions to the n+1 problem as well. Finally, we consider a separate class of solutions, namely travelling waves. The wave solutions we obtain here are distinct from those discussed previously in the literature. For these solutions, we are able to determine the asymptotic behavior of the solutions

Asymptotic Solutions For Singularly Perturbed Boussinesq Equations

We consider a family of singularly perturbed Boussinesq equations. We obtain a rational weak solution to the classical Boussinesq equation and demonstrate that this solution can be used to construct perturbation solutions for singularly perturbed high-order Boussinesq equations. These solutions take the form of an algebraic function which behaves similarly to a peakon, and which decays as time becomes large. We show that approximate solutions obtained via perturbation for the singularly perturbed models are asymptotic to the true solutions as the residual errors rapidly decay away from the origin. © 2012 Elsevier Inc. All rights reserved